A is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number.

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78 views

Where to learn about continued fractions?

I sought a convergent subsequence of $\sin n$, and I met a proof that utilizes continued fractions. I always stumble upon things related to them as well. So, I would like to learn about them. What is ...
5
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2answers
43 views

Can we define sum and product of two irrational numbers using Cauchy sequences of their simple continued fraction convergents?

There is a lot of questions about sum and product of irrationals here, so I hope you'll bear with me. Simple continued fraction is a very convenient way to represent any number since every real ...
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0answers
73 views

a continued fraction related to the exponential function $e^x$

Given a natural number $n$,with $|x|\lt1$ define the following conjectured identity $$G(n,x)=\begin{aligned}\cfrac{-n}{1-x-\cfrac{(1+n)(1-x^2)}{1-x^3-\cfrac{x^2(1-x)(1-x^3)}{1-x^5-\cfrac{x^3(1-x^2)(1-...
4
votes
1answer
60 views

While resolving a continued fraction by simplifying it into a quadratic equation, does the extraneous root have any significance?

Suppose we want to find the value of the the following expression, with infinite terms: $$ y = {1\over 1+ {1\over 1+ {1\over 1+ {1\over \ddots}}}} $$ To solve this, we follow the following procedure, ...
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0answers
38 views

What kinds of functions can be expanded into a form of an infinite continued fraction?

In this 1943 paper by Hudson on the theory of elastic waves in beams the author offers an interesting way to calculate a function defined as $$ \theta_n (z)=\frac{z J'_n (z)}{J_n (z)} $$ with $z$ ...
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2answers
162 views

the ratio of jacobi theta functions and a new conjectured q-continued fraction

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define $$\begin{aligned}H(q)=\cfrac{2(1+q^2)}{1-q+\cfrac{(1+q)(1+q^3)}{1-q^3+\cfrac{2q^2(1+q^4)}{1-q^5+\cfrac{q^3(1+q)(1+q^5)}{1-q^7+\cfrac{q^...
3
votes
1answer
69 views

Coloring rational numbers

Here is my problem. Fix a color for the number $1$, for example yellow. Choose another color, for example green. Now, for a positive rational denoted $x$, there are two rules : $x$ and $1/x$ have ...
7
votes
2answers
257 views

a conjecture of certain q-continued fractions

Given the squared nome $q=e^{2i\pi\tau}$ with $|q|\lt1$, define, $$\begin{aligned}F(q)=\cfrac{1-q^2}{1-q^3+\cfrac{q^3(1-q)(1-q^5)}{1-q^9+\cfrac{q^6(1-q^4)(1-q^8)}{1-q^{15}+\cfrac{q^9(1-q^7)(1-q^{11})}...
3
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0answers
101 views

conjectured identity of the product of two theta functions

Looking into the discussion in this post,I was naturally led to consider the following general identity Given the two jacobi theta functions,$$\theta_2(q)=\sum_{n=-\infty}^\infty q^{(n+1/2)^2}$$ and $...
0
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1answer
95 views

Determine convergents for square root

All square roots can be represented as a continued fraction. The fraction can be calculated to $n$ terms (e.g. $\sqrt{2}$ is $[1; 2, 2, 2, 2...]$) So the continued fraction for $\sqrt{2}$ to $3$ ...
5
votes
1answer
132 views

Rogers-Ramanujan continued fraction in terms of theta functions?

The Rogers-Ramanujan cfrac is, $$r = r(\tau)= \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\ddots}}}$$ If $q = \exp(2\pi i \tau)$, then it is known that, $$\frac{1}{r}-r =\frac{\eta(\tau/5)}{\eta(5\...
5
votes
2answers
194 views

a certain simple continued fraction

Given the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}$$ and the following simple continued fraction: $$G(q,k)=\cfrac{1}{1-q+\cfrac{1}{1-{q^3}^k+\cfrac{1}{1-{q^5}^k+\cfrac{1}{1-{q^7}^k+\ddots}}}}$$ For ...
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0answers
70 views

Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide

We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove ...
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1answer
44 views

Why is this continued fraction expansion what it is?

We have to find the continued fraction expansion of the roots of $1553 t^2 + 6014 t + 5820 = 0$, that is, $(\sqrt{14356} - 6014) / 3106$ Simplifying, $(\sqrt{3589}- 3007) / 1553$ The continued ...
6
votes
2answers
181 views

A $q$-continued fraction connected to the divisor function?

In this post, the following two continued fractions discussed by Nicco are given, $$A(q)= \left(\frac{\vartheta_2(0,q)}{2\,q^{1/4}}\right)^2= \cfrac{1}{1-q+\cfrac{q(1\color{red}-q)^2}{1-q^3+\cfrac{q^...
8
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1answer
126 views

are these two continued fractions equivalent?

I would like to pose the following conjecture.Given $$\phi(q) =\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q^3(1-q^2)^2}{1-q^5+\cfrac{q^5(1-q^3)^2}{1-q^7+\ddots}}}}$$ and $$\psi(q)=\cfrac{-q}{1-q+\...
3
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1answer
92 views

an interesting q-series and a certain continued-fraction

My aim is to find a rigorous proof of the following conjectured identity.Given $$1+q+q^2-q^4-q^5+q^7+q^8-q^{10}-q^{11}+\ddots=\cfrac{1}{1-q+\cfrac{(q^3)}{1-q^3+\cfrac{q^2(1+q)(1+q^3)}{1-q^5+\cfrac{q^...
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2answers
168 views

a conjectured new generating function of narayana's sequence

In the 14th century ,an Indian mathematician T.V Narayana came up with a sequence now named after him.The sequence satisfies the recurrence $$a_{n}=a_{n-1}+a_{n-3}$$ Starting with $a_{0}=a_{1}=1$, ...
5
votes
1answer
198 views

conjectured new generating function of fibonacci numbers

I conjecture a new generating function for the fibonacci numbers $F_{n}$. Given,the following conjectured q-continued fraction $$\chi(q)=\cfrac{1}{1+q-\cfrac{(1+q^2)}{1+q^3+\cfrac{q^2(1-q)(1-q^3)}{1+...
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0answers
41 views

Riemann Zeta continued fraction approximants

In the paper Continued-Fraction Expansions for the Riemann Zeta Function and Polylogarithms by Djurdje Cvijovic and Jacek Klinowski, there is a claim that I cannot reproduce. In the abstract they ...
4
votes
1answer
221 views

When $x=2$ in the infinitely continued fraction $x+\frac{1}{x^2+\frac{1}{x^3+\frac{1}{x^4+…}}}$, what algebraic value does it converge to?

Say you have the infinitely continued fraction: $$x+\frac{1}{x^2+\frac{1}{x^3+\frac{1}{x^4+...}}}$$ When $x=1$, you can see that it's $$1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}$$ which converges ...
3
votes
2answers
96 views

Why am I getting two answers for 8th root of continued fraction

Find value of $x$: $x=\sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-....and\,so\, on}}}$ On solving ,we have $x^8=2207-\frac{1}{x^8}$ $x^8+\frac{1}{x^8}=2207$ $x^4+\frac{1}{x^4}=47$ $x^2+\frac{1}{x^2}...
12
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0answers
147 views

Bi-linear relation between two continued fractions

We know that any positive real number $x$ can be represented as a simple continued fraction $$x = a_{0} + \dfrac{1}{a_{1} + \dfrac{1}{a_{2} + \dfrac{1}{a_{3} + \cdots}}} = [a_{0}, a_{1}, a_{2}, a_{3}, ...
7
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2answers
347 views

Ramanujan theta function and its continued fraction

I believe Ramanujan would have loved this kind of identity. After deriving the identity, I wanted to share it with the mathematical community. If it's well known, please inform me and give me some ...
7
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4answers
198 views

how to solve $3 - \cfrac{2}{3 - \cfrac {2}{3 - \cfrac {2}{3 - \cfrac {2}{…}}}}$

$$A = 3 - \cfrac{2}{3 - \cfrac {2}{3 - \cfrac {2}{3 - \cfrac {2}{...}}}}$$ My answer is: $$\begin{align} &A = 3 - \frac {2}{A}\\ \implies &\frac {A^2-3A+2}{A}=0\\ \implies &A^2-3A+2=0\\ \...
6
votes
3answers
97 views

Find the value of this infinite term

goes on till infinity. I get two solutions by rewriting the term in the form of the equation $x = 3-(2/x)$, which are $1$ and $2$. But in my opinion this term should have only one possible value. ...
2
votes
2answers
80 views

Continued Fraction Counting Problem

The house of my friend is in a long street, numbered on this side one, two, three, and so on. All the numbers on one side of him added up exactly the same as all the numbers on the other side of him. ...
18
votes
1answer
509 views

a new continued fraction for $\sqrt{2}$

In a q-continued fraction related to the octahedral group I defined a new q-continued fraction for the square of ramanujan's octic continued fraction which I discovered using certain three term ...
12
votes
1answer
133 views

Continued fraction estimation of error in Leibniz series for $\pi$.

The following arctan formula for $\pi$ is quite well known $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots\tag{1}$$ and bears the name of Madhava-Gregory-Leibniz series after ...
5
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2answers
252 views

The ratio of jacobi theta functions

Let $q=e^{2\pi i\tau}$. If $\theta_2$ and $\theta_3$ are jacobi theta functions , is it true that the ratio of the two functions can be expressed as a continued fraction of the form $$ \frac{\theta_2(...
3
votes
0answers
154 views

a q-continued fraction related to the octahedral group

Let $q=e^{2\pi i\tau}$. If $u(\tau)$ is Ramanujan's octic continued fraction, $$u(\tau)=\cfrac{\sqrt{2}\,q^{1/8}}{1+\cfrac{q}{1+q+\cfrac{q^2}{1+q^2+\cfrac{q^3}{1+q^3+\ddots}}}}$$ is it true that ...
9
votes
1answer
136 views

Are the unit partial quotients of $\pi, \log(2), \zeta(3) $ and other constants $all$ governed by $H=0.415\dots$?

Khinchin showed that given the simple continued fraction of a real number, $$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$ then it is almost always true that the partial quotients $a_i$ ...
5
votes
2answers
350 views

What is the geometric average of 2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10…?

It is known that the Khinchin constant is not the geometric mean of the first $n$ coeffecients, as $n$ approaches infinity, of the continued fraction of e, which is $$[2; 1, 2, 1, 1, 4, 1,1, 6, 1, 1, ...
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2answers
63 views

Visualisation of the reciprocal of an continued fraction?

If: $$a=\cfrac{l}{m+\cfrac{n}{o+\cfrac{p}{q+\cdots}}}$$ Then could you help me visualize $1/a$? I really don't understand it. Thank you so much!
3
votes
1answer
186 views

A new $q$-continued fraction of order $12$

I think I may have discovered a $q$-continued fraction of order $12$ with a form different from that established by Mahadeva Naika. Let $q=e^{2i \pi \tau}=\exp(2i \pi \tau)$, then, $$\begin{aligned} ...
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0answers
67 views

The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over 2}-{1+\...
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0answers
44 views

For which prime numbers $p$ there exist $x,y\in \Bbb{Z}$ such that $p=x^2+2y^2$? [duplicate]

For which prime numbers $p$ there exists $x,y\in \Bbb{Z}$ such that $p=x^2+2y^2$? I guess I am to use continued fraction, but I am not sure how. I know how to find solutions for defined numbers but I ...
2
votes
1answer
44 views

Computing periodic continued fractions.

Compute $[1,2,3,\overline{1,4}]$ where $\overline{1,4}$ is the periodic part. I looked into explanations about that, but haven't come by an actual algorithm of computing such a thing. I know it is ...
3
votes
1answer
138 views

A real number is rational $\iff$ its continued fraction expansion is finite.

I know that if this expansion is finite, then I can go to the lowest denominator in the whole fraction and turn it into a fraction and keep doing so until I get a fraction which means the number is ...
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2answers
80 views

How can I approximate a decimal with two fractions where denominator is less or equal to $d$

I was looking for a way to approximate a decimal number with a fraction, whose denominator is less or equal to $d$. Basically, having a decimal $X$, I want to find two fractions such that $$\frac{a_1}...
4
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0answers
84 views

Is there an advantage in using continued fractions for Catalan or Fibonacci-Lucas primality tests?

I am studying the basic theory about continued fractions and also reviewed here at MSE former questions to learn more. While reviewing the questions and answers, I found references to the Fibonacci ...
6
votes
1answer
96 views

Find the derivative of $f$ if it exists, else, prove it doesn't exist

Let $f: \mathbb R^+ \to \mathbb R $ $$f=\mathop{\vcenter{\LARGE\mathrm K}}\limits_{j=1}^{+\infty}\frac{x}{x^j}=\cfrac{x}{x^1+\cfrac{x}{x^2+\cfrac{x}{x^3+\ddots}}}$$ I saw this problem in a math ...
6
votes
1answer
122 views

Even Fibonacci Numbers and $\sqrt{5}$

My question is simple, but a mystery to me. Skip to the last paragraph if you're not interested in the story of my exploration. EDIT: I seem to have misinterpreted a key detail regarding how the ...
2
votes
2answers
120 views

Finding two solutions to $x^2 - 6y^2 = 1$ using continued fractions [closed]

Can anyone show me how to find the solutions to $x^2-6y^2=1$ by using continued fractions? I know how to find the fractions for $\sqrt6$ but do not know how to proceed. THANK YOU!!!
3
votes
0answers
119 views

Continued fraction for $\int_{0}^{\infty}(e^{-xt}/\cosh t)\,dt$

In one of the comments to a question I posted on MSE, I got this wonderful continued fraction $$\int_{0}^{\infty}\frac{e^{-xt}}{\cosh t}\,dt = \frac{1}{x +}\frac{1^{2}}{x +}\frac{2^{2}}{x +}\frac{3^{2}...
3
votes
0answers
118 views

Integral formulas involving continued fractions

Ramanujan posed the following formulas as questions in the Journal of Indian Mathematical Society: $$\int_{0}^{\infty}\dfrac{\sin nx\,\,dx}{{\displaystyle x + \dfrac{1}{x +}\dfrac{2}{x +}\dfrac{3}{x +...
0
votes
1answer
39 views

Farey Sequence implemenatation

I'm trying to use the Farey sequence to get the next lowest reduced fraction in a list. For example, for $n = 8$, we have $\dots, \frac13, \frac38, \frac25, \frac37, \frac12, \dots$ So let's take $\...
-1
votes
1answer
41 views

Continued Fraction Expansions Confusion

Let $\theta$ be an irrational number with continued fraction expansion $[a_0; a_1, a_2, \cdots]$. Suppose $P_n/Q_n = [a_0; a_1, \cdots , a_n]$ is the $n^{th}$ convergent. Then how do I show that $P_0=...
0
votes
2answers
71 views

Theorem 1 in Khinchin's “Continued Fractions”

I'm reading an English translation of Khinchin's Continued Fractions and I may have found an error in Theorem 1, page 4. Khinchin observes that if we simplify a finite continued fraction $[a_0; a_1, ....
8
votes
3answers
125 views

Minimal $ab$ for Rational Number $a/b$ in an Interval

Given rational numbers $L$ and $U$, $0<L<U<1$, find rational number $M=a/b$ such that $L \le M<U$ and $(a\times b)$ is as small as possible---$a$ and $b$ are integers. For example, If $...